UNIT 4: Building Polynomial and Rational Functions/Equations
Standards:
N.Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
N.Q.2 Define appropriate quantities for the purpose of descriptive modeling.
N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
F.IF.2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums;
symmetries; end behavior; and periodicity.*
F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*
c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.* Emphasize the selection of a model function based on behavior of data and context.
N.CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
A.APR.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
A.APR.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
A.REI.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
Essential Questions to answer during this unit:
1. How do I graph a polynomial function?
2. How do I determine an appropriate domain for a polynomial model given data?
3. How do I find the roots of a polynomial function?
4. How does the Remainder Theorem help me to find the roots of a polynomial function?
5. How do I determine the equation and graph of a polynomial function given its roots?
6. How do I solve a rational equation?
7. How do I use rational equations to solve problems?
You must answer all these questions by the end of the Unit and bring in real world applications of polynomial functions and rational equations
Vocabulary:
asymptote, continuity, depressed polynomial, end behavior, point of discontinuity, rational equation, rational function, relative maximum, relative minimum, synthetic substitution, vertical asymptote, remainder theorem, fundamental theorem of Algebra.
What will we study?
- Polynomial Functions
- Graphing polynomial Functions
- Solving Equations Using Quadratic Techniques
- Revisiting the Reminder and Factor Theorem
- Roots and Zeros
- Solving Rational Equations and Inequalities
Polynomial Functions - A polynomial equation used to represent a function.
eg. 1) f(x) = x
Defn: A polynomial function, f, of degree n is a function of the form:
F(x) = a₀ xⁿ +….. a ( a₀ ≠ 0 )
*** Note: degree is the highest exponent nonnegative integer
***Note: a is a constant term (term without a variable)
Constant Function has degree zero. eg.: ______
Graph is a horizontal line.
Linear Function has degree one. eg.:______
Graph is a slanted line.
Quadratic Function has degree two. eg.______
Graph is a parabola.
y = ax
when a > 0, “happy” parabola eg.______
when a < 0, “sad” parabola
eg.______
axis of symmetry (folding line)
x =
vertex (turning point) (maximum for sad, minimum for happy)
x value is the same as axis of symmetry. Substitute back into y = equation to get y-value.
The end behavior: is the behavior of the graph as x is approaching + ∞ or -∞
The graph of even degree function may or may not intersect the x-axes depending on its location in coordinate plan. If intersects the x-axes in 2 places, the function has two real zeros.
What happens when the graph does not intersect x axes?
______
What happens if the graph is tangent with x-axes?
______
The graph of an odd degree function always crosses the x-axes at least once, and thus the function always has at least one real zero.
To sketch a polynomial function:· Note the degree of the polynomial
-- use it to predict the general shape and end behavior.
-- even functions "start high & end high"
-- odd functions "start low & end high"
· Note the coefficient of the term with highest degree
-- use it to determine if the curve is reflected about the x-axis.
x³ "starts low and ends high"
(generally increases as x increases)
- x³ "starts high and ends low"
(generally decreases as x increases)
· Rewrite it by factoring
identify the linear, quadratic, or other factors.
· Plot the real zeros.
· Note for each root or zero what kind of a root it is
-- odd powers pass through the x-axis,
-- even powers touch but do not pass through the x-axis.
· Plot (0, f(0)), the y-intercept.
· Solve: first derivative = 0
-- to find relative maximums/minimums.
· Determine sign in intervals
-- using the positiveness or negativeness of each factor.
-- just "connect the dots."
· Sketch curve.